Nonlinear force-free coronal magnetic field extrapolation scheme based on the direct boundary integral formulation

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL.???, XXXX, DOI: /, Nonlinear force-free coronal magnetic field extrapolation scheme based on the direct boundary integral formulation Han He, 1,2 and Huaning Wang 1 Abstract. The boundary integral equation (BIE) method was first proposed by Yan and Sakurai [2000] and used to extrapolate the nonlinear force-free magnetic field in the solar atmosphere. Recently, Yan and Li [2006] improved the BIE method and proposed the direct boundary integral equation (DBIE) formulation, which represents the nonlinear force-free magnetic field by direct integration of the magnetic field on the bottom boundary surface. Based on this new method, we devised a practical calculation scheme for the nonlinear force-free field extrapolation above solar active regions. The code of the scheme was tested by the analytical solutions of Low and Lou [1990] and was applied to the observed vector magnetogram of solar active region NOAA The results of the calculations show that the improvement of the new computational scheme to the scheme of Yan and Li [2006] is significant, and the force-free and divergence-free constraints are well satisfied in the extrapolated fields. The calculated field lines for NOAA 9077 present the X-shaped structure, and can be helpful for understanding the magnetic configuration of the filament channel as well as the magnetic reconnection process during the Bastille Day flare on 14 July Introduction Both observations and theoretical analyses reveal that the magnetic field plays an important role in the activity phenomena of the solar atmosphere: moving plasmas are confined to magnetic field lines, and the magnetic field also provides the energy for solar flares and other eruptive phenomena [Tsuneta et al., 1992; Masuda et al., 1994; Shibata et al., 1995; Wang et al., 1996; Tsuneta, 1996; Priest and Forbes, 2002; Shibata, 2004; Lin et al., 2005; Schwenn, 2006]. To understand the physical mechanisms of these activities in the solar atmosphere, an important step is to find out the underlying structure of the magnetic field above the related active region. Currently, the direct measurement of the magnetic field in the solar chromosphere and corona is not as sophisticated as observation in the photosphere. The commonly used method for understanding the configurations of the magnetic fields above the photosphere is extrapolation: the fields can be reconstructed from a physical model in which the observed photospheric magnetic field is taken as a boundary condition. The force-free field model is often adopted for this purpose, since it is a reasonable approximation in the solar chromosphere and corona [Metcalf et al., 1995]. The nonlinear force-free magnetic field, with the field lines being everywhere aligned parallel to the electric current density, can be described by equations: ( B) B = 0, (1) B = 0. (2) Equation (2) is the divergence-free constraint of the magnetic field. The force-free constraint (1) can also be written 1 National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China. 2 Kwasan and Hida Observatories, Graduate School of Science, Kyoto University, Kyoto, Japan. Copyright 2008 by the American Geophysical Union /08/$ as B = α(r)b, (3) where α is a function of spatial location, usually called the force-free parameter or force-free factor. It is constant along each field line, which can be determined from the bottom boundary condition. Currently, several methods for the nonlinear force-free field (NLFFF) extrapolation have been proposed [Sakurai, 1981; Yan and Sakurai, 2000; Wheatland et al., 2000; Wiegelmann, 2004; Régnier and Amari, 2004; Valori et al., 2005; Yan, 2005; Wiegelmann et al., 2006; Yan and Li, 2006; Amari et al., 2006; Wheatland, 2006; Song et al., 2006; Schrijver et al., 2006; Wiegelmann, 2007; Song et al., 2007; Valori et al., 2007; Fuhrmann et al., 2007]. As one of the techniques for nonlinear force-free field modeling, the boundary integral equation (BIE) method was first proposed by Yan and Sakurai [2000]. Recently, Yan and Li [2006] improved the BIE and proposed the direct boundary integral equation (DBIE) formulation, which represents the force-free magnetic field by direct integration of the magnetic field on the bottom boundary surface. Figure 1 shows the geometry for application of DBIE [Yan and Li, 2006]. On the bottom boundary surface Γ (infinite plane), the boundary condition is B = B 0 on Γ, (4) where B 0 denotes the known boundary values which can be supplied from vector magnetogram measurements. At infinity, an asymptotic constraint condition is also introduced to ensure a finite energy content in the semispace Ω above Γ, B = O(r 2 ) when r, (5) where r is the radial distance. According to the DBIE method, after a series of derivations using the two constraint conditions (2)-(3) and the two boundary conditions (4)-(5), the magnetic strength at the field point (x i, y i, z i) in Ω can be represented by the equation [Yan and Li, 2006]: B i = Γ Y n Γ B0dΓ = Y B0dΓ. (6) z

2 X - 2 The reference function Y in equation (6) is chosen as Y = cos(λρ) 4πρ cos(λρ ) 4πρ, (7) where ρ = [(x x i) 2 + (y y i) 2 + (z z i) 2 ] 1/2 is the distance between a variable point (x, y, z) and the given field point (x i, y i, z i), ρ = [(x x i) 2 +(y y i) 2 +(z +z i) 2 ] 1/2, as shown in Figure 1. The parameter λ in equation (7) has the same dimension (reciprocal of length) as the force-free factor α, which is defined by equation Y ( λ 2 B α 2 B α B ) dω = 0. (8) Ω In the case of a nonlinear force-free field, corresponding to B ix, B iy, and B iz in equation (6), there exist three components of the reference function, Y x, Y y, and Y z, together with λ x, λ y, and λ z defined locally at the given field point by equation (8). The values of the three components of λ are generally different [Li et al., 2004; Yan and Li, 2006; He and Wang, 2006]. Once the parameter λ x, λ y, and λ z at the field point (x i, y i, z i) are given, the magnetic field B i can be calculated by integration of the magnetic field on the bottom boundary surface through the DBIE formulation (6). Because it is costly to determine λ directly from the volume integral equation (8), Yan and Li [2006] devised an optimal approach to find the suitable λ values locally by using the DBIE (6) together with the force-free constraint condition (1). At the field point (x i, y i, z i), they defined an evaluate function as: f i(λ x, λ y, λ z) = J B, with J = B, (9) J B which measures the absolute value of sine of the angle between J and B. For any initial values of λ x, λ y, and λ z, B at the field point and its neighborhood can be calculated by DBIE (6), then the value of f i(λ x, λ y, λ z) can be obtained by equation (9). In an ideal situation, the suitable λ values (denoted by λ x, λ y, and λ z) can be determined by f i(λ x, λ y, λ z) = 0, (10) which is equivalent to the force-free field equation (1). In practical computing, one needs to find (λ x, λ y, λ z) that satisfies f i(λ x, λ y, λ z) = min f i(λ x, λ y, λ z). (11) A downhill simplex method can be employed to fulfil the task of equation (11) [Nelder and Mead, 1965; Yan and Li, 2006]. The computation of Yan and Li [2006] shows that, if equation (10) is satisfied, the divergence-free constraint (2) can also be satisfied at the neighborhood of the given field point. This property indicates that the divergence-free constraint (2) has been involved during the derivation of the DBIE formulation (6). We followed the main idea of the strategy proposed by Yan and Li [2006], and devised the upward boundary integration scheme for the nonlinear force-free field extrapolation. That is, while we take into account the whole boundary data information through the DBIE formulation (6), the parameter λ at a given field point can be determined locally through the force-free equation (1) or equation (10) with the help of only the neighboring boundary data information. Unlike the original procedure of Yan and Li [2006], our computation is carried out upwardly layer by layer. The procedure and techniques of the new scheme is described in section 2. The code of the scheme was tested by the analytical solutions of Low and Lou [1990] and was applied to the observed vector magnetogram of solar active region NOAA The results of test calculations using the analytical solutions are presented in section 3, and the results of the calculations by using the observed vector magnetogram are shown in section 4. In section 5, we give the summary and conclusion. 2. Upward Boundary Integration Scheme In the original computational procedure proposed by Yan and Li [2006], the equation (11) has better convergence property at the field points with lower altitude (explained in Appendix A). To take advantage of this property and to avoid the increasing errors at the field points with higher altitude, we devised a new scheme for nonlinear force-free field extrapolation based on the DBIE formulation (6), which we called the upward boundary integration scheme. In the new scheme, as shown in Figure 2, the computation is carried out upwardly, from Γ 0 (photosphere) to Γ 1, then from Γ 1 to Γ 2, and so on. In the original procedure of Yan and Li [2006], the bottom boundary was fixed to the photosphere, while in our scheme the bottom boundary for applying the DBIE formulation (6) is moved upwardly layer by layer. That is, we always calculate the field distribution at Γ n+1 from the data of the new bottom boundary Γ n. The step distance between the two consecutive layers is the same as the space between the two consecutive grid points at Γ n (the scale of one pixel). Since Γ n+1 is very close to Γ n, the field points at Γ n+1 are always at very low altitude relative to the bottom boundary Γ n. Then equation (11) can be used to achieve good convergence at every layer. In some circumstances, especially in the case of J or B approaches to zero in equation (9), equation (11) may fail to achieve a convergent result. The problem appears in the form of isolated points where singularities appear in the raw data of Γ n+1, which can be eliminated by smoothing processes with the help of the nearest grid points in the x- and y- directions. In the original procedure of Yan and Li [2006], the values of f i and J in equation (9) are calculated in the infinitesimal neighborhood ±δ of the field point in the x-, y-, and z- directions (small cubic volume surrounding the field point). The computing of the integration in equation (6) should be done seven times (six sides of the small cube plus the field point itself) to obtain the value of f i and J. To reduce the loads of computing and fully utilize the boundary data information at Γ n in our scheme, we calculate f i and J in a small square pyramid between the field point i and the neighboring grid points at the boundary Γ n, as shown in Figure 3. The height of the pyramid is l (distance between Γ n+1 and Γ n) and the side length of the square base is 2l, where l denotes the space between two consecutive grid points at Γ n (the scale of one pixel). Given an arbitrary λ at field point i, to obtain the values of f i(λ) and J(λ) in equation (9) we only need to carry out the integration of DBIE (6) one time at the field point i to calculate B i(λ), thus saving computing time as compared to the original procedure of Yan and Li [2006]. Once B i(λ) is known, J x, J y, and J z can be calculated in two isosceles triangles and the square base of the pyramid, respectively, as indicated by different colors in Figure 3. B o at the center of the square base, as shown in Figure 3, is employed to complete the calculation of f i(λ). Then we can use equation (11) to find the suitable value of λ. Once λ is determined, B i(λ ) will be the final result at the field point i for the nonlinear force-free field modeling. We employed the same code of the downhill simplex method as used by Yan and Li [2006] to perform multidimensional minimization of f i in equation (11). The initial starting point of (λ x, λ y, λ z) for the downhill simplex

3 X - 3 method is selected as (0, 0, 0) in our code. The characteristic length scale of λ in the downhill simplex method is specified as the maximum absolute value of force-free factor α at the boundary surface Γ 0, because λ and α have the same dimension (reciprocal of length) and the same order of magnitude [Li et al., 2004; Yan and Li, 2006; He and Wang, 2006]. For Case I and Case II in section 3 (analytical solutions of Low and Lou [1990]), the maximum absolute values of α are 11.3 and 11.1, respectively. If we adopt one pixel as the length unit (with a grid for boundary area x, y [ 1, +1], see section 3) as in our code, the two values become and For the data of active regions observed in the photosphere (see section 4), we choose the characteristic length scale of λ as m 1 [Pevtsov et al., 1995]. Also in our code, the length unit is one pixel, for the common active regions with a field of view and pixel number 64 64, the characteristic length scale of λ becomes 0.342, which approximates to the values used by Case I and Case II of the analytical solutions. As described in section 1, the DBIE formulation (6) demands that the bottom boundary Γ be an infinite plane surface [Yan and Li, 2006]. Figures 4 and 5 illustrates how to apply the DBIE to a solar active region with concentrated magnetic flux. At the bottom boundary Γ, the observed vector magnetogram is bounded in a finite square area which covers the main magnetic flux of the active region as shown in Figure 4. The remained flux outside the square area is relatively very weak, and can be considered to be zero as an approximation approach. Then, in practical calculation, we only need to carry out the integration of DBIE (6) over the finite square area of the active region, while the bottom boundary Γ is still an infinite plane surface. Considering that the main magnetic flux region as well as the field lines may expand at higher layers, the square area for the integration is enlarged gradually layer by layer during the calculation as illustrated in Figure 5 (in present code, from Γ n to Γ n+1, each side of the square area expanding by one pixel). Meanwhile, we keep the original pixel number of the square area at all layers by resampling the grid points to save computing time. Thus the space between the two consecutive grid points at Γ n as well as the distance between the two consecutive layers also increases gradually with height, as shown in Figure 5. After the field distributions at a series of layers are obtained, the values of the magnetic field in the space between the layers can be calculated through the technique of interpolation. In practical calculation, the code reads the boundary data (square area, N N array in the code), calculates the nonlinear force-free field distributions at all layers, and re-forms the grid structure to a regular form through interpolation. The output of the code is the field distribution in a cubic volume (N N N array in the code) which is just above the boundary data area, as illustrated in Figures 4 and 5. The current code is written in IDL programming language. The time needed for a output grid is about 13 hours on an 1.86GHz Intel processor. 3. Testing the Code by Using the Analytical Solutions of Low and Lou [1990] First, we test the code described in section 2 by using the analytical nonlinear force-free field solutions given by Low and Lou [1990]. The fields of the analytical solutions are basically axially symmetric. The point source of the axisymmetric fields is located at the origin of the spherical coordinate system, with the axis of symmetry pointing to the Z direction associated with the spherical coordinate system. By arbitrarily positioning the plane surface boundary Γ of DBIE in the space of the analytical fields, we obtain a different kind of boundary conditions prepared for extrapolation. Two cases are selected to test the validity and accuracy of our code as shown in Figure 6, which are the same cases as used by Schrijver et al. [2006]. The parameters for Case I are n = 1, m = 1, L = 0.3, Φ = π/4, the parameters for Case II are n = 3, m = 1, L = 0.3, Φ = 4π/5, where n and m are the eigenvalues of the solutions, L is the distance between the plane surface boundary Γ and the point source (origin of the spherical coordinate system), and Φ is the angle between the normal direction of Γ and the Z axis associated with the spherical coordinate system [Low and Lou, 1990; Schrijver et al., 2006]. In both cases, the field distributions in the modeling volume bounded by x, y [ 1, +1] and z [0, 2] (x, y, and z are Cartesian coordinates defined locally on the boundary surface Γ) were calculated based on the bottom boundary data in the area x, y [ 1, +1], just as described in section 2 and illustrated in Figures 2-5. The pixel numbers of the bottom boundary data are Direct Comparison The extrapolated field lines for Case I and Case II are compared with the analytical solutions in Figures 7 and 8, respectively. Left columns are images of analytical solutions, right columns are images of calculated fields. It can be seen from the figures that the orientations of the extrapolated field lines basically coincide with the analytical solutions. Figures 7 and 8 only show the inner volume (x, y [ 0.5, +0.5] and z [0, 1], with grid) of the modeling space as indicated in Figure 6 by square dashed lines, where the quality of agreement between the extrapolated field and the analytical solutions is better than in the margin region. A vector correlation metric C vec is employed to quantify the degree of agreement between the analytical field B and the extrapolated field b as used by Schrijver et al. [2006]. To define C vec, the equation Bi bi i C vec = ( i Bi 2 bi 2), (12) 1/2 i with B i and b i as the field vectors of the analytical field B and the extrapolated field b at each grid point i. If B and b are identical, C vec = 1; if B i b i, C vec = 0. We calculated the values of C vec at each layer in the central domain (x, y [ 0.5, +0.5] and z [0, 2], with grid) of the modeling space, the results are shown in Figure 9. Since we only use the finite boundary data in the area x, y [ 1, +1] with grid, it can be seen that the extrapolated fields b deviate from the analytical fields B gradually with the increasing of height. At the lower layers and in the central domain, B and b get the best agreement, as illustrated in Figures 7 and Internal Consistency of the Extrapolated Field The internal consistency of the calculated field is measured by the force-free constraint (1) and divergence-free constraint (2). To check the extent to which the extrapolated fields satisfy the force-free and divergence-free constraints, we introduced the integral measures L f of the Lorentz force and L d of divergence of the fields, as used by Schrijver et al. [2006]. L f and L d are defined as: L f = 1 V V L d = 1 V B 2 ( B) B 2 dv, (13) V B 2 dv. (14) We calculated L f and L d of the extrapolated field at each layer in the central domain (x, y [ 0.5, +0.5] and z [0, 2],

4 X - 4 with grid, the unit of length in the calculation is one pixel) and plotted the curves of L f vs height and L d vs height. The results for Case I and Case II are shown in Figures 10 and 11, respectively. The curves for analytical solutions of Low and Lou [1990] are also calculated and shown in Figures 10 and 11 for reference. The curves for the analytical solutions (Figures 10a and 10c, Figures 11a and 11c) show a typical profile of L f vs height and L d vs height for the nonlinear force-free field. That is, the measures L f and L d decrease to zero rapidly with the increase of height. The relatively large values of L f and L d at the lower layers are the effects of the discrete grid points and the finite-difference method used in the calculations, and the values of L f and L d at the bottom layer (layer number 0) are in the same order of magnitude (Exact values of L f and L d at the bottom layer can be found in Tables 1 and 2.). The curves for the extrapolated field (Figures 10b and 10d, Figures 11b and 11d) show similar properties of the profile as do the analytical solutions, which indicates that the force-free constraint (1) and divergence-free constraint (2) are well satisfied in the extrapolated field. Comparing the analytical solutions, the larger values of L f and L d for the extrapolated field are the results of the errors existing in the calculated fields, which were introduced by the numerical computation and are amplified by the B 2 terms in equations (13) and (14). Besides the values at each layer, the integral measures L f and L d were also calculated over the entire volume (x, y [ 1, +1] and z [0, 2], with grid) as well as the inner volume (x, y [ 0.5, +0.5] and z [0, 1], with grid). All the values of L f and L d for Case I and Case II are listed in Tables 1 and 2, respectively. The maximum values of B at the bottom boundary are also included in Tables 1 and 2 for reference Comparing with Original Computational Scheme of Yan and Li [2006] As discussed in section 2 and Appendix A, in the original computational scheme of Yan and Li [2006], the equation (11) has better convergence property at the field points with lower altitude (near the bottom boundary) than the field points with higher altitude (away from the bottom boundary). To take advantage of this property and avoid the increasing errors at the field points with higher altitude, we devised the new upward boundary integration scheme for NLFFF extrapolation. To check the improvement of the new scheme to the original computational procedure of Yan and Li [2006], we produced the curves of C vec vs height for Case I by using the original computational scheme proposed by Yan and Li [2006] and by using the new upward boundary integration scheme. The two curves are compared in Figure 12. The curve of C vec vs height measures the agreement between the extrapolated field and the analytical solution at each layer. C vec = 1 represents the best agreement as explained in section 3.1. It can be seen from the curves in Figure 12 that the extrapolated field from the original scheme of Yan and Li [2006] basically agrees with the analytical solution at lower layers, but at higher layers, the two fields are totally different. The extrapolated field by the new scheme is still similar to the analytical field at higher layers, as shown in Figure 12. The extrapolated field lines of the new scheme (Figure 7d) compared with the field lines of the original scheme (Figure 18a) also illustrate this improvement. Several techniques are introduced in the code of the new extrapolation scheme to save computing time as described in section 2. To calculate the NLFFF in a volume with a grid for Case I, the current code of the new scheme needs about 13 hours on an 1.86GHz Intel processor in IDL programming language, while the code of the original scheme of Yan and Li [2006] (also in IDL programming language) needs about 104 hours on the same computer Comparing with Other Integration Scheme for NLFFF Extrapolation Five evaluation metrics C ves, C CS, E n, E m and ɛ are introduced by Schrijver et al. [2006] to quantify the degree of agreement between the analytical fields B and the extrapolated field b in the modeling volume. The first metric C ves are defined in equation (12), the other four metrics are defined as [Schrijver et al., 2006]: C CS = 1 M i B i b i B i b i, (15) E n bi Bi i = 1 i Bi, (16) E m = 1 1 M i b i B i, (17) B i i ɛ = bi 2, (18) Bi 2 i where B i and b i are the field vectors of the analytical field B and the extrapolated field b at each grid point i, M is the total number of vectors in the volume. If B and b are identical, all of the five metrics equal one. To have a quantitative comparison to other integration schemes and other NLFFF extrapolation methods, we calculated the five metrics for the extrapolated field of the new computational scheme in the entire volume (x, y [ 1, +1] and z [0, 2], with grid), inner volume (x, y [ 0.5, +0.5] and z [0, 1], with grid), and the lower central domain (x, y [ 0.5, +0.5] and z [0, 0.5], with grid). The results for Case I and Case II are listed in Tables 3 and 4, respectively. The values of the integral scheme implemented by Liu in the paper of Schrijver et al. [2006] and the original computational scheme proposed by Yan and Li [2006] (only for lower central domain) are also included in Tables 3 and 4 for reference. It should be noted that the boundary conditions in our calculations are somewhat different to the boundary conditions used by Schrijver et al. [2006]. In the work of Schrijver et al. [2006], the data on all six boundaries of the modeling volume are available for Case I, the data on the bottom boundary area x, y [ 3, +3] (with grid) are provided for Case II. In our calculations, we use the bottom boundary data in the area x, y [ 1, +1] (with grid) for both Case I and Case II. Tables 3 and 4 show that the degree of agreement of the new scheme is better than the integral scheme implemented by Liu [Schrijver et al., 2006]. 4. Applying the Code to the Observed Vector Magnetogram of Solar Active Region In this section, we check the validity and compatibility of the extrapolation code to deal with the observed boundary data of solar active regions. The vector magnetogram employed for testing was observed by Solar Magnetic Field Telescope (SMFT) [Ai, 1987], which is located at Huairou Solar Observing Station of NAOC (National Astronomical Observatories, Chinese Academy of Sciences) in Beijing. The active region associated with the magnetogram is NOAA The data were observed at 04:14 UT on 14 July 2000, several hours before the Bastille Day event (X5.7 flare) at 10:24UT [Deng et al., 2001; Yan et al., 2001; Liu

5 X - 5 and Zhang, 2001; Zhang et al., 2001; Zhang, 2002; Tian et al., 2002; Somov et al., 2002; Wang et al., 2005]. The original field of view of the magnetogram is from a pixel size CCD. We reformed the magnetogram to a square area ( ) through cropping and interpolation and reduced the pixel number to (4.2 /pixel). The final magnetogram which is ready for the NLFFF extrapolation is shown in Figure 13a. The extrapolated field lines are shown in Figures 13b-13d. The lines in blue color are closed field lines with both footpoints being anchored at the bottom boundary, the red lines representing the field lines that leave the modeling box. Figure 13b is the top view of the field configuration in the modeling volume above the whole magnetogram. Figures 13c and 13d show the detailed structures above the central domain of the magnetogram (32x32 grid, indicated by a square dashed line in Figure 13a). Figure 13c is the top view of the field lines, Figure 13d is the 3-D view of the field lines along the direction of the polarity inversion line as indicated by an arrow in the right margin of Figure 13c. All the field lines in Figures 13b-13d are plotted from layer 1 (see Figure 17) to avoid the influence of noises in the data of the bottom boundary (the magnetogram observed in the photosphere). In Figures 13b and 13c, we can see the compact loops with different orientations aligned over the polarity inversion line of the magnetogram. The 3-D view of the field lines in Figure 13d shows that the loops above the right half of the polarity inversion line in Figure 13c is lower than the loops above the left half of the polarity inversion line. Figure 13d also shows that, in the region with lower arcade (right half of Figure 13c), the open field lines together with the underlying compact loops present an X-shaped structure. The diagram to illustrate this X-shaped structure is sketched in Figure 14e. The dashed curve in Figure 14e represents the U-shaped field lines above the X-point, which are not plotted and thus are displayed as a cavity in Figure 13d. The whole 3-D views of the closed field lines and the U-shaped field lines above the main polarity inversion line of the magnetogram are shown in Figures 14a and 14b, the side views of the field lines along the direction of Y-Axis are shown in Figures 14c and 14d, respectively. To check the extent to which the extrapolated field can reflect the real distributions of the coronal magnetic fields, We compared the extrapolated field lines of NOAA 9077 with the EUV images of solar atmosphere obtained by the Transition Region and Coronal Explorer (TRACE) satellite [Handy et al., 1999]. The top views of the closed field lines and the U-shaped field lines are plotted in Figure 15, and are compared with the TRACE 195Å image of the same region at almost the same time (04:12 UT on 14 July 2000). It can be seen from Figures 15e and 15f that the orientations of the closed field lines above the left half of the polarity inversion line basically coincide with the coronal loops observed in the TRACE 195Å image, while in the region above the right half of the polarity inversion line, Figures 15c and 15d show that the distribution of the U-shaped field lines basically coincides with the configuration of the filament channel. To check the extent to which the extrapolated fields of the active region NOAA 9077 satisfy the force-free constraint (1) and divergence-free constraint (2), we calculated the integral measure L f and L d at each layer of the calculated field, and produced the curves of L f vs height and L d vs height, as we have done for the analytical field in section 3. The two curves are shown in Figure 16. The profiles of the curves in Figure 16 show similar properties to the analytical solutions in section 3. That is, the measures L f and L d decrease to zero rapidly with the increase in height, indicating that the force-free and divergence-free constraints are well satisfied in the extrapolated field. Being suitable for dealing with the noisy vector magnetogram observations is one of the great advantages of the BIE/DBIE method [Yan and Sakurai, 2000; Yan, 2005; Yan and Li, 2006]. The integration over the whole boundary in the DBIE formulation (6) can efficiently suppress the influence of the noises in the boundary data to the convergence property of equation (11). Thus the extrapolation code for the analytical solutions in section 3 can be applied to the real vector magnetograms directly. Figure 17 shows the magnitude distribution of the extrapolated field for NOAA 9077 at four successive layers. Figure 17a is the magnitude distribution of B at the bottom boundary (layer 0), Figures 17b-17d are the magnitude distribution at layers 1 to 3. Since the field are calculated layer by layer, the noises in the boundary data are eliminated quickly with the increase in height. 5. Summary and Conclusion Based on DBIE formulation (6), we devised the upward boundary integration scheme for the nonlinear force-free field extrapolation. In this new scheme, the bottom boundary for applying the DBIE formulation (6) is moved upwardly layer by layer. That is, we always calculate the field distribution at Γ n+1 from the data of the new bottom boundary Γ n, as shown in Figure 2. While we take into account the whole bottom boundary data information at Γ n through the DBIE formulation (6), the suitable value of parameter λ at a given field point in Γ n+1 can be determined locally through the force-free constraint condition (11) with the help of only neighboring boundary data information at Γ n, as shown in Figures 2 and 3. The main techniques employed in the new scheme include: (a) the bottom boundary for applying the DBIE is moved upwardly layer by layer to achieve the best convergence property and accuracy, as shown in Figure 2; (b) the parameter λ at a given field point is calculated in a small square pyramid (sketched in Figure 3) to fully utilize the boundary data information at Γ n, and thus save the computing time; (c) the square area for computing the integration of DBIE (6) is enlarged gradually layer by layer to fit the expanding field, at the same time, pixel number of the square areas is fixed at all layers by resampling the grid points to save the computing time as illustrated in Figure 5. The code of the new computational scheme was tested by the analytical solutions of Low and Lou [1990] and are applied to the observed vector magnetogram of solar active region. In the direct comparison between the extrapolated fields with the analytical solutions of Low and Lou [1990], the orientations of the extrapolated field lines are basically coincide with the analytical solutions. The quantitative comparison shows that the extrapolated fields deviate from the analytical fields gradually with the increase in height. At the lower layers and in the central domain, the best agreements are obtained. Since we only use the finite bottom boundary data in the calculation, and the analytical solutions of Low and Lou [1990] present global configurations [Low and Lou, 1990; Wang and Sakurai, 1998; Li et al., 2004; He and Wang, 2006], it is natural that the extrapolated fields deviate from the analytical fields at higher layers. The force-free constraint (1) and divergence-free constraint (2) for the nonlinear force-free field are well satisfied in the extrapolated field as discussed in section 3.2. In the original computational scheme of Yan and Li [2006], the equation (11) has better convergence property at the field points with lower altitude (near the bottom boundary) than the field points with higher altitude (away from the bottom boundary). The new upward boundary integration scheme is proposed for taking advantage of this property and to avoid the increasing errors at higher altitude. The improvement of the new scheme to the original computational procedure of Yan and Li [2006] is significant as shown

6 X - 6 in section 3.3. The accuracy of the new scheme is better than the integral scheme implemented by Liu [Schrijver et al., 2006] as discussed in section 3.4. If we only care about the magnetic field near the bottom boundary (at layers with very low altitude), the integral scheme of Liu can also give reasonable results as shown in Tables 3 and 4. The calculations using the observed vector magnetogram of solar active region NOAA 9077 in section 4 demonstrate that the DBIE formulation (6) and the upward boundary integration scheme can be applied to solar active regions for the nonlinear force-free field extrapolation. Only the bottom boundary data in the photosphere are needed in the calculation, and the DBIE method can suppress the noises in the observed data through the integration over the whole bottom boundary. The force-free and divergence-free constraints are well satisfied in the extrapolated field as shown in Figure 16. Since the computation is carried out point by point and layer by layer as described in section 2, fine structures can be preserved in the extrapolated field of NOAA 9077 as illustrated in Figures In the region above the left half of the polarity inversion line of the magnetogram, the orientations of the closed field lines basically coincide with the coronal loops observed in the TRACE 195Å image as shown in Figures 15e and 15f. In the region with the dark filament above the right half of the polarity inversion line, the extrapolated field lines present the X-shaped structure as demonstrated in Figures Beneath the X-point are the lowlying compact loops along the polarity inversion line, while above the X-point, the distribution of the U-shaped field lines coincides with the configuration of the filament channel as shown in Figures 15c and 15d, which can be helpful for understanding the magnetic structure of the dark filament as well as the magnetic reconnection process during the Bastille Day flare [Tsuneta et al., 1992; Masuda et al., 1994; Shibata et al., 1995; Tsuneta, 1996; Masuda et al., 2001; Fletcher and Hudson, 2001; Priest and Forbes, 2002; Somov et al., 2002; Shibata, 2004]. The application of the new DBIE extrapolation scheme to the real vector magnetograms is still in a preliminary stage. More efforts on the comparison between the extrapolated field lines and the coronal loop observations are needed. The recent data obtained by Hinode (Solar-B) satellite are very valuable for this purpose [Kosugi et al., 2007]. Topological methods and techniques will be useful for quantitatively analyzing the topological properties of the extrapolated fields [Wang et al., 2000, 2001; Longcope, 2005; Zhao et al., 2005]. Moreover, the DBIE formulation can be applied to the spherical boundary case [Aly and Seehafer, 1993; Li et al., 2004; Yan, 2005; He and Wang, 2006]. The upward boundary integration scheme can also be adapted to the case with a spherical boundary. By using the full disk observations of vector magnetograms [UeNo et al., 2004; Zhang et al., 2007; Su and Zhang, 2007], it is possible to model the global or large-scale structures of a coronal magnetic field with a spherical boundary, and study their relationships to the coronal mass ejections (CMEs) [Chen and Shibata, 2000; Chen et al., 2002; Zhang and Low, 2005; Zhou et al., 2006a, b]. Appendix A: Convergence Property of the Original Computational Procedure Proposed by Yan and Li [2006] We use the analytical nonlinear force-free field solutions of Low and Lou [1990] to investigate the convergence property of the original computational procedure proposed by Yan and Li [2006]. The boundary conditions and boundary data are the same as Case I described in section 3. The field distribution in the space above the bottom boundary is calculated by using the original DBIE extrapolation code provided by Yan and Li ( [Yan and Li, 2006]. The extrapolated field lines in the inner volume (x, y [ 0.5, +0.5] and z [0, 1], with grid) are shown in Figure 18. The values of the extrapolated field are smoothed with the help of nearest grid points before the field lines are plotted. Figure 18a is the 3-D view, Figure 18b is the side view along the direction of Y-Axis. It can be seen from Figure 18 that the orientations of the field lines at lower altitude basically coincide with the analytical solution (Figure 7c). But the field lines at higher altitude are contorted and lose regularity. This result indicates that, in the original computational procedure of Yan and Li [2006], the equation (11), employed to find the suitable λ values for DBIE (6), has a better convergence property at the field points with lower altitude (near the bottom boundary) than the field points with higher altitude (away from the bottom boundary). To take advantage of this property and avoid the increasing errors at the field points with higher altitude, we devised our new scheme for nonlinear force-free field extrapolation based on the DBIE formulation (6). In the new scheme, the values of magnetic field are calculated layer by layer upwardly as described in section 2. Acknowledgments. The authors thank the colleagues for valuable comments and discussions during the CAWSES International Workshop on Space Weather Modeling (CSWM) at the Earth Simulator Center in Yokohama. We are grateful to the reviewers for valuable comments and suggestions to improve the paper, and Dr. Clia Goodwin for great help to improve the readability of the manuscript. We thank Prof. Yihua Yan and Dr. Zhuheng Li for helpful discussions on the principle of the BIE/DBIE method and the techniques of force-free field extrapolation. H. 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8 X - 8 Zhou, G. P., Y. M. Wang, and J. X. Wang (2006b), Coronal mass ejections associated with polar crown filaments, Advances in Space Research, 38, H. He, and H. N. Wang, National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing , China. (hehan@bao.ac.cn; hnwang@bao.ac.cn)

9 X - 9 Table 1. Integral Measures L f and L d of the Extrapolated Field for Case I a Analytical Solution Extrapolated Field L f over the inner volume b L d over the inner volume L f over the entire volume c L d over the entire volume L f at the bottom layer of inner volume L d at the bottom layer of inner volume L f at the bottom layer of entire volume L d at the bottom layer of entire volume Maximum of B at the bottom boundary a L f and L d are defined in equations (13) and (14), and the unit of length in the calculations is one pixel. The non-zero values of L f and L d for the analytical solution are due to the discrete grid points and the finite-difference method used in the calculations. b Inner volume is bounded by x, y [ 0.5, +0.5] and z [0, 1], with grid. c Entire volume is bounded by x, y [ 1.0, +1.0] and z [0, 2], with grid. Table 2. Integral Measures L f and L d of the Extrapolated Field for Case II a Analytical Solution Extrapolated Field L f over the inner volume b L d over the inner volume L f over the entire volume c L d over the entire volume L f at the bottom layer of inner volume L d at the bottom layer of inner volume L f at the bottom layer of entire volume L d at the bottom layer of entire volume Maximum of B at the bottom boundary a L f and L d are defined in equations (13) and (14), and the unit of length in the calculations is one pixel. The non-zero values of L f and L d for the analytical solution are due to the discrete grid points and the finite-difference method used in the calculations. b Inner volume is bounded by x, y [ 0.5, +0.5] and z [0, 1], with grid. c Entire volume is bounded by x, y [ 1.0, +1.0] and z [0, 2], with grid. Table 3. Evaluation Metrics for Case I a C ves C CS E n E m ɛ New scheme of DBIE (entire volume b ) New scheme of DBIE (inner volume c ) New scheme of DBIE (lower central domain d ) Original scheme of DBIE (lower central domain) Integral scheme implemented by Liu (entire volume) e Integral scheme implemented by Liu (inner volume) Integral scheme implemented by Liu (lower central domain) a The five evaluation metrics are defined in equations (12) and (15)-(18). In the new and original computational schemes of DBIE, only bottom boundary data (x, y [ 1, +1] with grid) were used for extrapolation, while in the scheme of Liu, data of the bottom boundary and the four side boundaries of the modeling volume were used. b Entire volume is bounded by x, y [ 1, +1] and z [0, 2], with grid. c Inner volume is bounded by x, y [ 0.5, +0.5] and z [0, 1], with grid. d Lower central domain is bounded by x, y [ 0.5, +0.5] and z [0, 0.5], with grid. e The values of the integral scheme implemented by Liu are recalculated by courtesy of Dr. Yang Liu. The small differences between the values listed here and the values in the paper of Schrijver et al. [2006] are due to the different accuracies of the numerical solutions for the analytical field.

10 X - 10 Table 4. Evaluation Metrics for Case II a C ves C CS E n E m ɛ New scheme of DBIE (entire volume b ) New scheme of DBIE (inner volume c ) New scheme of DBIE (lower central domain d ) Original scheme of DBIE (lower central domain) Integral scheme implemented by Liu (entire volume) e Integral scheme implemented by Liu (inner volume) Integral scheme implemented by Liu (lower central domain) a The five evaluation metrics are defined in equations (12) and (15)-(18). In the new and original computational schemes of DBIE, data of the bottom boundary area x, y [ 1, +1] (64 64 grid) were used for extrapolation, while in the scheme of Liu, data of the bottom boundary area x, y [ 3, +3] ( grid) were used. b Entire volume is bounded by x, y [ 1, +1] and z [0, 2], with grid. c Inner volume is bounded by x, y [ 0.5, +0.5] and z [0, 1], with grid. d Lower central domain is bounded by x, y [ 0.5, +0.5] and z [0, 0.5], with grid. e The values of the integral scheme implemented by Liu are recalculated by courtesy of Dr. Yang Liu. The small differences between the values listed here and the values in the paper of Schrijver et al. [2006] are due to the different accuracies of the numerical solutions for the analytical field.

11 X - 11 B = O( r z 2 ) x Γ : Ω : B = α B B = 0 ( x, y, z) B = B 0 n ( i i i x, y, z ) ρ ρ ( i i i x, y, z ) y Figure 1. The geometry for application of the direct boundary integral equation (DBIE). boundary Γ + n 1 Γ 4 Γ 3 boundary Γ n Γ 2 Γ 1 Γ 0 Figure 2. The diagram to illustrate the upward boundary integration scheme for nonlinear force-free field extrapolation based on the DBIE formulation (6). The computation is carried out layer by layer upwardly, the field distribution at Γ n+1 is calculated from the data of Γ n. J x field point i J y J z z y B o x l Γ n Figure 3. The diagram to illustrate the calculation of J( B) in the small square pyramid between the field point i and the neighboring grid points at Γ n. The height of the square pyramid is l and the side length of the square base is 2l, where l denotes the space between two consecutive grid points at Γ n. J x and J y are calculated in two isosceles triangles respectively, J z is calculated in the square base of the pyramid, as indicated by different colors.

12 X - 12 B = O( r 2 ) Y Bi = B 0 dγ z Γ active region infinite plane surface boundary Γ Figure 4. The diagram to illustrate the boundary conditions for application of the DBIE method to a solar active region with concentrated magnetic flux. At the bottom boundary Γ, the observed vector magnetogram is bounded in a finite square area which covers the main magnetic flux of the active region. The remained flux outside the square area is relatively very weak, and can be considered to be zero as an approximation approach. The cubic volume above the square boundary data area represents the output region of the code. Γ active region Layer 5 Layer 4 Layer 3 Layer 2 Layer 1 Layer 0 Figure 5. The diagram to illustrate how to fit expanding main magnetic flux region as well as field lines at higher layers in the new computational scheme for the DBIE method. The area for the integration of the DBIE is enlarged gradually layer by layer. Meanwhile, pixel number of the areas is fixed at all layers by resampling the grid points to save the computing time. The square dashed line indicates the final output region of the code.

13 X - 13 (a) Case I z x Γ (b) Case II x z Γ Figure 6. The global field configurations of the two analytical NLFFF solutions given by Low and Lou [1990], which are employed to test the validity and accuracy of the NLFFF extrapolation code (see section 3). The parameters for Case I are n = 1, m = 1, L = 0.3, Φ = π/4, the parameters for Case II are n = 3, m = 1, L = 0.3, Φ = 4π/5, where n and m are the eigenvalues of the solutions, L is the distance between the plane surface boundary Γ and the point source (origin of the spherical coordinate system), and Φ is the angle between the normal direction of Γ and the Z axis associated with the spherical coordinate system [Low and Lou, 1990; Schrijver et al., 2006]. The radius of the spheres is equal to L(=0.3). The long thick lines indicate the position and scale of the bottom boundary data area (x, y [ 1, +1]). The square dashed lines represent the inner volume (x, y [ 0.5, +0.5] and z [0, 1]) of the modeling space. x, y, and z are Cartesian coordinates defined locally on the plane surface boundary Γ.

14 X - 14 (a) (b) (c) (d) Figure 7. The extrapolated field lines in the inner volume (x, y [ 0.5, +0.5] and z [0, 1]) of the modeling space for Case I compared with the analytical solution of Low and Lou [1990]. Left column (a) and (c) are images of the analytical solution, right column (b) and (d) are images of the extrapolated field. Top row are images in top view, bottom row are images in 3-D view. Closed field lines are plotted in blue, field lines that leave the modeling box are in red.

15 X - 15 (a) (b) (c) (d) Figure 8. The extrapolated field lines in the inner volume (x, y [ 0.5, +0.5] and z [0, 1]) of the modeling space for Case II compared with the analytical solution of Low and Lou [1990]. Left column (a) and (c) are images of the analytical solution, right column (b) and (d) are images of the extrapolated field. Top row are images in top view, bottom row are images in 3-D view. Closed field lines are plotted in blue, field lines that leave the modeling box are in red.

16 X - 16 Figure 9. The curves of vector correlation metric C vec (defined in equation (12)) vs height for Case I (dotted line) and Case II (dashed line). The extrapolated fields are calculated by the code of the new computational scheme. C vec is calculated at each layer in the central domain x, y [ 0.5, +0.5] and z [0, 2], with grid. (a) analytical solution vs height L f (b) extrapolated field vs height L f Case I Case I (c) analytical solution L vs height d (d) extrapolated field vs height L d Case I Case I Figure 10. L f vs height and L d vs height curves of the extrapolated field by using the new scheme for Case I compared with the analytical solution of Low and Lou [1990]. Left column (a) and (c) are curves of the analytical solution, right column (b) and (d) are curves of the extrapolated field. L f and L d are integral measures of the Lorentz force and divergence, as defined in equations (13) and (14). L f and L d are calculated at each layer in the central domain (x, y [ 0.5, +0.5] and z [0, 2], with grid), the unit of length in the calculation is one pixel.

17 X - 17 (a) analytical solution vs height L f (b) extrapolated field vs height L f Case II Case II (c) analytical solution L vs height d (d) extrapolated field vs height L d Case II Case II Figure 11. L f vs height and L d vs height curves of the extrapolated field by using the new scheme for Case II compared with the analytical solution of Low and Lou [1990]. Left column (a) and (c) are curves of the analytical solution, right column (b) and (d) are curves of the extrapolated field. L f and L d are integral measures of the Lorentz force and divergence, as defined in equations (13) and (14). L f and L d are calculated at each layer in the central domain (x, y [ 0.5, +0.5] and z [0, 2], with grid), the unit of length in the calculation is one pixel.

18 X - 18 Figure 12. The curve of C vec vs height for Case I by using the original computational scheme proposed by Yan and Li [2006] (dashed line), compared with the curve produced by using the new upward boundary integration scheme (dotted line). C vec is defined in equation (12) and is calculated at each layer in the inner volume (x, y [ 0.5, +0.5] and z [0, 1], with grid).

19 X - 19 (a) (b) d (c) (d) Figure 13. Results of the NLFFF extrapolation for active region NOAA 9077 by using the code of the new scheme. (a) The vector magnetogram used by the code. The white contours represent the positive polarity of B z, the black contours represent the negative polarity of B z, the contour levels are ±50, 100, 200, 500, 1000, 1500, 2000, 3000G. Small arrows overlying the contours represent B t (transverse component). The data was observed at 04:14 UT on 14 July 2000 by the Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station. The field of view is with pixel number as (4.2 /pixel). The square dashed line indicates the central domain (32 32 grid) of the magnetogram. (b) The top view of the extrapolated field. (c) The top view of the extrapolated field in the central domain. (d) The 3-D view of the extrapolated field in the central domain along the direction of the polarity inversion line as indicated by an arrow in the right margin of Figure 13c. Closed field lines are plotted in blue color, field lines that leave the modeling box are in red color. All field lines are plotted from layer 1 (see Figure 17) to avoid the influence of noises in the data of bottom boundary.

20 X - 20 ( a) (b) ( c) ( d) (e) S N Figure 14. (a)(b) The whole 3-D views of the closed field lines and the U-shaped field lines in the extrapolated field of NOAA 9077 (see section 4 and Figure 14e). The closed field lines are plotted from layer 1 (see Figure 17) to avoid the influence of noises in the data of bottom boundary. The U-shaped lines are plotted in the volume between layer 1 and layer 19. (c)(d) The side views of the closed field lines and the U-shaped field lines along the direction of Y-Axis. (e) Diagram to illustrate the X-shaped structure of the field lines in the extrapolated field of NOAA 9077 as shown in Figure 13d. The dashed curve represents the U-shaped field lines above the X- point, which are not plotted and thus are displayed as a cavity in Figure 13d.

21 (a ) ( b) (c) (d ) (e) (f ) Figure 15. Comparison between the extrapolated field lines of NOAA 9077 and the TRACE 195A image. (a) The vector magnetogram of the selected region which covers the main polarity inversion line of NOAA The data were observed at 04:14 UT on 14 July (b) The TRACE 195A image of the same region. The data were observed at 04:12 UT on 14 July (c) The U-shaped field lines overlying the magnetogram of Bz. (d) The U-shaped field lines overlying the TRACE 195A image. (e) The closed field lines overlying the magnetogram of Bz. (f) The closed field lines overlying the TRACE 195A image. The U-shaped lines in Figures 15c and 15d are plotted in the volume between layer 0 and layer 19. The closed field lines in Figures 15e and 15f are plotted from layer 0 (see Figure 17). X - 21

22 X - 22 ( a) vs height NOAA 9077 L f ( b) vs height NOAA 9077 L d Figure 16. L f vs height and L d vs height curves of the extrapolated fields by using the code of the new scheme for NOAA L f and L d are defined in equations (13) and (14) and were calculated at each layer. The unit of magnetic field is Gauss; the unit of length in the calculation is one pixel.

23 X - 23 (a) (b) ( c) (d) Figure 17. Magnitude distribution of the extrapolated field by using the code of the new scheme for NOAA 9077 at four successive layers. (a) The magnitude distribution of B at the bottom boundary (layer 0); (b)-(d) The magnitude distributions of B at layers 1 to 3. The unit of magnetic field (Z-Axis) is Gauss. (a) (b) Figure 18. The extrapolated field lines in the inner volume (x, y [ 0.5, +0.5] and z [0, 1]) of the modeling space for Case I by using the original computational procedure proposed by Yan and Li [2006]. (a) The image in 3-D view. (b) The image in side view along the direction of Y-Axis. Closed field lines are plotted in blue, field lines that leave the modeling box are in red.